US9830555B2 - Computation using a network of optical parametric oscillators - Google Patents

Computation using a network of optical parametric oscillators Download PDF

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US9830555B2
US9830555B2 US14/904,040 US201414904040A US9830555B2 US 9830555 B2 US9830555 B2 US 9830555B2 US 201414904040 A US201414904040 A US 201414904040A US 9830555 B2 US9830555 B2 US 9830555B2
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opo
computational
phase
optical
machine
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US20160162798A1 (en
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Alireza Marandi
Yoshihisa Yamamoto
Robert Byer
Zhe Wang
Shoko Utsunomiya
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Inter-University Research Institute Corp Research Organization Of Information
Inter University Research Institute Corp Research Organization of Information and Systems
Leland Stanford Junior University
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    • G06N99/002—
    • G—PHYSICS
    • G02—OPTICS
    • G02F—OPTICAL DEVICES OR ARRANGEMENTS FOR THE CONTROL OF LIGHT BY MODIFICATION OF THE OPTICAL PROPERTIES OF THE MEDIA OF THE ELEMENTS INVOLVED THEREIN; NON-LINEAR OPTICS; FREQUENCY-CHANGING OF LIGHT; OPTICAL LOGIC ELEMENTS; OPTICAL ANALOGUE/DIGITAL CONVERTERS
    • G02F3/00—Optical logic elements; Optical bistable devices
    • G—PHYSICS
    • G06—COMPUTING OR CALCULATING; COUNTING
    • G06E—OPTICAL COMPUTING DEVICES; COMPUTING DEVICES USING OTHER RADIATIONS WITH SIMILAR PROPERTIES
    • G06E3/00—Devices not provided for in group G06E1/00, e.g. for processing analogue or hybrid data
    • G06E3/001—Analogue devices in which mathematical operations are carried out with the aid of optical or electro-optical elements
    • G06E3/005—Analogue devices in which mathematical operations are carried out with the aid of optical or electro-optical elements using electro-optical or opto-electronic means
    • G—PHYSICS
    • G06—COMPUTING OR CALCULATING; COUNTING
    • G06N—COMPUTING ARRANGEMENTS BASED ON SPECIFIC COMPUTATIONAL MODELS
    • G06N3/00—Computing arrangements based on biological models
    • G06N3/02—Neural networks
    • G06N3/04—Architecture, e.g. interconnection topology
    • G06N3/044—Recurrent networks, e.g. Hopfield networks
    • G—PHYSICS
    • G06—COMPUTING OR CALCULATING; COUNTING
    • G06N—COMPUTING ARRANGEMENTS BASED ON SPECIFIC COMPUTATIONAL MODELS
    • G06N3/00—Computing arrangements based on biological models
    • G06N3/02—Neural networks
    • G06N3/04—Architecture, e.g. interconnection topology
    • G06N3/047—Probabilistic or stochastic networks
    • G—PHYSICS
    • G06—COMPUTING OR CALCULATING; COUNTING
    • G06N—COMPUTING ARRANGEMENTS BASED ON SPECIFIC COMPUTATIONAL MODELS
    • G06N3/00—Computing arrangements based on biological models
    • G06N3/02—Neural networks
    • G06N3/06—Physical realisation, i.e. hardware implementation of neural networks, neurons or parts of neurons
    • G06N3/067—Physical realisation, i.e. hardware implementation of neural networks, neurons or parts of neurons using optical means
    • G06N3/0675—Physical realisation, i.e. hardware implementation of neural networks, neurons or parts of neurons using optical means using electro-optical, acousto-optical or opto-electronic means
    • G—PHYSICS
    • G06—COMPUTING OR CALCULATING; COUNTING
    • G06N—COMPUTING ARRANGEMENTS BASED ON SPECIFIC COMPUTATIONAL MODELS
    • G06N5/00—Computing arrangements using knowledge-based models
    • G06N5/01—Dynamic search techniques; Heuristics; Dynamic trees; Branch-and-bound
    • G—PHYSICS
    • G02—OPTICS
    • G02F—OPTICAL DEVICES OR ARRANGEMENTS FOR THE CONTROL OF LIGHT BY MODIFICATION OF THE OPTICAL PROPERTIES OF THE MEDIA OF THE ELEMENTS INVOLVED THEREIN; NON-LINEAR OPTICS; FREQUENCY-CHANGING OF LIGHT; OPTICAL LOGIC ELEMENTS; OPTICAL ANALOGUE/DIGITAL CONVERTERS
    • G02F1/00—Devices or arrangements for the control of the intensity, colour, phase, polarisation or direction of light arriving from an independent light source, e.g. switching, gating or modulating; Non-linear optics
    • G02F1/35—Non-linear optics
    • G02F1/39—Non-linear optics for parametric generation or amplification of light, infrared or ultraviolet waves

Definitions

  • NP nondeterministic polynomial time
  • a computational machine includes an optical device configured to receive energy from an optical energy source and generate a number N 1 of optical signals, and a number N 2 of coupling devices, each of which controllably couples a plurality of the number N 1 optical signals.
  • the coupling devices are individually controlled to simulate a computational problem.
  • a computational machine includes a number N 1 of parametric oscillators and a number N 2 of coupling devices, each of which controllably couples a plurality of the number N 1 of parametric oscillators together.
  • the coupling devices are individually controlled to simulate a computational problem.
  • OPOs optical parametric oscillators
  • FIG. 3 illustrates an example of optical source pump pulses and corresponding OPO output signal pulses.
  • FIGS. 7A, 7B illustrate a numerical simulation of finding a solution using a computational machine.
  • FIGS. 7C, 7D illustrate a numerical simulation of finding a solution using a computational machine.
  • FIGS. 7E, 7F illustrate a numerical simulation of finding a solution using a computational machine.
  • FIG. 8A illustrates a numerical estimation of success probability's dependence on coupling strength between degenerate OPOs.
  • FIG. 8B illustrates a numerical estimation of success probability of finding ground states of the Ising Hamiltonian as a function of the number of spins.
  • FIG. 9 illustrates numerical estimation of computation time dependence on the number of spins.
  • FIG. 10A illustrates below threshold and above threshold behavior of an OPO.
  • FIG. 10B illustrates gradual pumping of the OPO network to find ground states, and illustrates that different spin configurations (phase states) have different total photon losses in a network.
  • FIG. 10C illustrates that the spin configuration corresponding to the ground state of the Ising problem is expected to oscillate above threshold.
  • FIG. 11A illustrates an experimental setup
  • FIG. 11B illustrates that the ring resonator of FIG. 11A has a roundtrip time of four times the pump pulse repetition period.
  • FIG. 11C illustrates coupling provided by the delays in the setup of FIG. 11A .
  • FIG. 12A illustrates some test results with all delay lines blocked in the setup of FIG. 11A .
  • FIG. 12B illustrates some test results with one delay line unblocked in the setup of FIG. 11A .
  • FIG. 12C illustrates some test results with all delay lines unblocked in the setup of FIG. 11A .
  • FIG. 13A illustrates some test results for the setup of FIG. 11A .
  • FIG. 13B illustrates a histogram of phase states when all couplings are blocked.
  • FIG. 13C illustrates a histogram of phase states when all couplings are unblocked and the phases are locked.
  • FIG. 14 illustrates a comparison of classical and quantum mechanical annealing techniques to the technique described in this disclosure.
  • FIG. 15 illustrates normalized build-up time
  • FIG. 16 illustrates average computation time of an example OPO network in solving MAX-CUT problems.
  • FIG. 17A illustrates an output spectrum
  • FIG. 17B illustrates an interferometric autocorrelation trace.
  • FIG. 17C illustrates a spatial beam profile
  • FIGS. 18A, 18B, 18C illustrate the output of a slow detector.
  • FIGS. 19A, 19B, 19C illustrate the output of a slow detector.
  • FIG. 20 illustrates an optical computation machine in a fiber-based implementation.
  • FIG. 21 illustrates a system according to this disclosure.
  • a computational machine as described in this disclosure finds absolute or approximate solutions for computational problems, including nondeterministic polynomial time computational problems.
  • the computational machine finds solutions for computational problems in the complexity class of “nondeterministic polynomial time,” known as “NP” in computational complexity theory, and further finds solutions for NP problems for which no polynomial-time technique is available.
  • NP nondeterministic polynomial time
  • a subset of NP problems is NP-complete problems. An NP problem may be mapped to an NP-complete problem with a polynomial time algorithm.
  • the Ising problem is to find a lowest-energy configuration, or ground state, for the Ising model.
  • An Ising model represents N sites in a magnetic structure, where a site represents a magnetic dipole, each site is coupled to a number of other sites, and each site has a spin state of either +1 or ⁇ 1. As the number of sites N and the number of couplings between the sites increases, the complexity of finding the ground states of the model increases exponentially, representing an NP-complete problem.
  • the computational machine described in this disclosure provides solutions for NP problems.
  • the NP-complete Ising problem is used as a non-limiting example in this disclosure by way of illustration.
  • the computational machine provides solutions for other NP and NP-complete problems also.
  • another NP-complete problem that may be solved using the computational machine is the MAX-CUT problem from graph theory.
  • Other problems include analyses of large social networks, and combinatorial optimization problems.
  • An Ising problem is explained as including a number N of sites and a matrix J of coupling terms between sites.
  • N the computational complexity of the problem exceeds the capability of existing supercomputers to find all solutions for all problems in finite time.
  • OPOs optical parametric oscillators
  • a network of N substantially identical OPOs with mutual coupling are pumped with the same source to simulate an Ising spin system.
  • the computational machine converges to one of the ground states and stays in it.
  • the phase state selection process depends on the vacuum fluctuations and mutual coupling of the OPOs.
  • the pump is pulsed at a constant amplitude, in other implementations the pump output is gradually increased, and in yet further implementations, the pump is controlled in other ways.
  • the computational machine includes a number N 1 of degenerate OPOs and a number N 2 of configurable couplings used for coupling the optical fields between OPOs.
  • the configurable couplings may be configured to be off, or configured to be on. Turning the couplings on and off may be performed gradually or abruptly. When configured to be on, the configuration may provide any phase or amplitude depending on the coupling matrix (J) of the Ising problem.
  • the computational machine as configured for the Ising problem uses a number M 1 of the N 1 degenerate OPOs and configures a number M 2 of the couplings to be on in a certain state. For some of the Ising problems, each of the M 2 couplings are configured to be in the same state. For other problems, the couplings may be configured to be in different states.
  • a pump source provides pump energy to three substantially identical degenerate OPOs 110 , 115 , 120 (labeled as OPO 1 , OPO 2 , and OPO 3 , respectively.) Pump energy may be provided as pulses or continuously.
  • the three OPOs are coupled to each other according to a coupling matrix J. As illustrated in FIG.
  • OPO 1 is coupled to OPO 2 with a coupling amplitude and phase identified in matrix element J 1,2
  • OPO 2 is coupled to OPO 3 with a coupling amplitude and phase identified in matrix element J 2,3
  • OPO 3 is coupled to OPO 1 with a coupling amplitude and phase identified in matrix element J 3,1 .
  • One example Ising problem is where the coupling amplitudes are substantially equal, and each coupling phase is equal to pi ( ⁇ ) .
  • each degenerate OPO When uncoupled, each degenerate OPO produces an output that randomly has either zero phase or ⁇ phase.
  • the network of OPO 1 , OPO 2 , and OPO 3 settles into a ground state with non-random outputs, where each output 111 , 116 , 121 , respectively, is at either zero phase or pi phase.
  • Each OPO output is combined with a phase reference 112 , 117 , or 122 , and the result is captured at a photodetector.
  • the OPO outputs 111 , 116 , 121 represent the Ising model in a ground state. For example, a zero phase may represent a ⁇ 1 spin state, and a ⁇ phase may represent a +1 spin state in the Ising model.
  • a zero phase may represent a ⁇ 1 spin state
  • a ⁇ phase may represent a +1 spin state in the Ising model.
  • the OPO network is implemented using a triplet OPO 205 , in which a resonant cavity of the triplet OPO is configured to have a round-trip time equal to three times the period of pulses from a pump source.
  • the pump source is shown as laser 210 and clock 215 .
  • Round-trip time as used herein indicates the time for light to propagate along one pass of a described recursive path.
  • the round-trip time for the triplet OPO of FIG. 2 indicates the time for light to propagate from entry to the triplet OPO at reflector M 1 , through reflectors M 2 , M 3 , and M 4 , and back to reflector M 1 .
  • Three pulses of a pulse train with period P equal to one-third of the resonator cavity round-trip time may propagate through triplet OPO 205 of FIG. 2 concurrently without interfering with each other. Therefore, three independent pulse trains can act as three temporally separated OPOs, which are illustrated in FIG. 2 as waves labeled OPO 1 , OPO 2 , and OPO 3 .
  • Two delay lines, namely delays A and B, are used for mutual coupling of the OPOs, and the modulators 220 , 225 synchronously control the strengths and phases of couplings, allowing for programming of the computational machine illustrated to simulate a desired problem.
  • Delay A is for coupling (OPO 1 to OPO 2 ), (OPO 2 to OPO 3 ), and (OPO 3 to OPO 1 )
  • Delay B is for coupling (OPO 1 to OPO 3 ), (OPO 2 to OPO 1 ), and (OPO 3 to OPO 2 ).
  • the phase states of the OPOs can be measured by interfering the output of triplet OPO 205 and a phase reference 230 .
  • FIG. 3 illustrates for the system in FIG. 2 an example in which pulse trains are provided by the pump source.
  • the state into which the system converges e.g., the simulation of an Ising model ground state
  • OPO 1 has phase zero
  • OPO 2 has phase ⁇
  • OPO 3 has phase zero.
  • the arrows on the signal pulses depict how the temporally separated OPOs are mutually coupled using the delay lines.
  • the round-trip time of the resonator cavity may be increased so as to be R times the repetition period T R of the pump pulses.
  • a number R ⁇ 1 of modulators and delay lines may be used to control amplitude and phase of coupling between OPOs.
  • FIG. 4 illustrates an example of a computational machine 400 for determining solutions for a number of sites N.
  • an unequal arm Michelson interferometer 410 may be used as shown. Each OPO pulse is interfered with the next pulse, and therefore the relative phase states of consecutive OPOs are measured.
  • a computational machine in accordance with this disclosure may be implemented using optical fiber technologies, such as fiber technologies developed for telecommunications applications.
  • optical fiber technologies such as fiber technologies developed for telecommunications applications.
  • fs femto-second
  • ps picosecond
  • ⁇ s microsecond
  • the couplings can be realized with fibers and controlled by optical Kerr shutters.
  • a computational machine in accordance with this disclosure may be implemented using miniaturized OPOs, which can be implemented in Planar Lightwave Circuits (PLCs) or using micro and nano resonators.
  • PLCs Planar Lightwave Circuits
  • FIG. 6 illustrates the results from operating the computational machine of FIG. 5 with no coupling (top trace), with in-phase coupling (middle trace), and with out-of-phase coupling (bottom trace).
  • each OPO output randomly takes one of the states zero or ⁇ , represented by the random switching between a higher voltage or a lower voltage, as measured by a photodetector.
  • both OPOs take the same phase state, which is represented by a higher voltage as measured by a photodetector.
  • the OPOs take opposite phase states and cancel each other, which is represented by the lower voltage as measured by a photodetector, where the lower voltage includes system noise.
  • the Ising problem is to find the ground states of the Ising Hamiltonian H (equation (1)) of a magnetic system consisting of N spins, where each spin ⁇ j takes either +1 or ⁇ 1, and J jl denotes the coupling between the j-th and the l-th spin.
  • Degenerate OPOs are open dissipative systems that experience second order phase transition at the oscillation threshold. Because of the phase-sensitive amplification, a degenerate OPO could oscillate with a phase of either 0 or ⁇ with respect to the pump phase for amplitudes above the threshold. The phase is random, affected by the quantum noise associated in optical parametric down conversion during the oscillation build-up. Therefore, a degenerate OPO naturally represents a binary digit specified by its output phase. Based on this property, a degenerate OPO system may be utilized as an Ising machine. The phase of each degenerate OPO is identified as an Ising spin, with its amplitude and phase determined by the strength and the sign of the Ising coupling between relevant spins. In the following, a theoretical investigation of the system using a set of classical equations is provided.
  • H H 0 + H int + H irr ( 2 )
  • H 0 ⁇ ⁇ ⁇ ⁇ s ⁇ a ⁇ s ⁇ ⁇ a ⁇ s + ⁇ p ⁇ a ⁇ p ⁇ ⁇ a ⁇ p ( 3 )
  • H int i ⁇ ⁇ ⁇ ⁇ ⁇ 2 ⁇ ( a ⁇ s ⁇ 2 ⁇ a ⁇ p - a ⁇ s 2 ⁇ a ⁇ p ⁇ ) + i ⁇ ⁇ ⁇ ⁇ ⁇ p ⁇ ( a ⁇ p ⁇ ⁇ F p ⁇ e - i ⁇ ⁇ ⁇ p ⁇ t - a ⁇ p ⁇ F p ⁇ e i ⁇ ⁇ ⁇ p ⁇ t ) ( 4 )
  • H irr i ⁇ ⁇ ⁇ ⁇ s ⁇ ( a ⁇ s
  • H int describes the nonlinear coupling between the signal and the pump fields, where ⁇ is the parametric gain due to the second order susceptibility of the nonlinear crystal; the second term shows the excitation of the internal pump field by the external pump.
  • H irr denotes the irreversible interaction between cavity fields and the reservoir, where ⁇ circumflex over (B) ⁇ s , ⁇ circumflex over (B) ⁇ p are reservoir operators, and ⁇ s , ⁇ p are the signal and the pump photon decay rates, respectively, in the cavity.
  • ⁇ circumflex over (f) ⁇ s , ⁇ circumflex over (f) ⁇ p are the time dependent noise operators to the signal and the pump fields, respectively.
  • Possible solution outputs from the network can be obtained by solving the c-number Langevin equations with the signal field of each degenerate OPO starting from the vacuum state.
  • quantum noise inputs in these equations makes the computational cost of this technique relatively expensive.
  • An alternative technique ignores the quantum noise terms and replaces them with a random initial condition.
  • the differential equations to be dealt with switch from stochastic ones to deterministic ones, which considerably improves the efficiency of the numerical simulation.
  • Equation 8A simplifies calculation by neglecting the noise term, as described above.
  • noise terms may be added to equation (8A), as shown in equation (8B).
  • the classical dynamic equations for a system of N coupled degenerate OPOs can be obtained by further including the mutual coupling among the degenerate OPOs.
  • the signal amplitude ⁇ j of the j-th degenerate OPO couples to that of the l-th degenerate OPO by 1 ⁇ 2 ⁇ jl , then the behavior of the network can be described by a set of N nonlinear differential equations, as shown in (9A) neglecting noise, or in (9B) with a noise term.
  • the system can also be equivalently described by the dynamics of the real number in-phase components C j and the quadrature components S j of the complex amplitudes ⁇ j .
  • normalized dynamic equations for the in-phase and quadrature components are utilized (e.g., equations (10), (11)), where
  • F th ⁇ s ⁇ ⁇ p 4 ⁇ ⁇ is threshold pump flux
  • c j C j /A s
  • Equation (17) is equation (13), in which the j-th quadrature component is multiplied by s j , and summed over all the equations.
  • a j 1+p+c j 2 , j ⁇ 1, 2, . . . , N ⁇ .
  • the last two terms of equation (17) are in the quadratic form of the matrix, as in (18).
  • equation (16) and equation (21) establish guidelines for choosing the appropriate combination of the normalized pump rate p and the coupling strength ⁇ jl to allow each degenerate OPO to represent a binary digit in the network. Yet, it is noteworthy that because of the sufficient nature of equation (21), there are exceptions to these conditions.
  • the 0-th order term c j (0) ⁇ square root over (p ⁇ 1) ⁇ is the signal amplitude of the j-th degenerate OPO at above the threshold when there is no mutual coupling.
  • the overall photon decay rate is, then, as shown in (24).
  • FIGS. 7A, 7B illustrate how a system, starting from random initial phases, finds a ground state of two spins with ferromagnetic coupling in the Ising model. However, when the system has more than one stable steady state, the system would approach to one of them based upon the initial conditions.
  • FIGS. 7C, 7D show that for some initial phases the system evolves to a steady state corresponding to a ground state of the Ising Hamiltonian.
  • FIGS. 7E, 7F show that for some other initial phases the system evolves to a steady state corresponding to an excited state of the Ising Hamiltonian.
  • the amplitudes of the degenerate OPOs are larger in the former case, as expected.
  • the statistics of the steady states approached by the system starting from 100 groups of 100 pairs of random initial phases are used to calculate the success probability and the standard deviation.
  • the success probability in this example is always above 0.5.
  • the ground state spin configuration does not change if the coupling strengths are scaled by the same positive factor. Thus, the efficiency of the system to find a ground state can be improved by scaling ⁇ properly.
  • the performance of coupled degenerate OPO networks as Ising machines can be evaluated on a restricted set of instances in the Ising model. In these instances, each spin couples to three others with anti-ferromagnetic coupling of identical strength ⁇ . Finding the ground state of these instances is equivalent to findING the maximum cut on unweighted cubic graphs.
  • the Ising Hamiltonian in the above equation is minimized when the weight of cut S is maximized.
  • the MAX-CUT problem restricted to unweighted cubic graphs is still NP-hard.
  • the spin configurations of the Ising model with N spins are obtained from the steady states of the corresponding coupled degenerate OPO network by solving the classical dynamic equations (10), (11) under different initial conditions.
  • the initial normalized amplitudes of the degenerate OPOs are set to be 10 ⁇ 5 , whereas the initial phases are randomly chosen as N different numbers to simulate the quantum noise associated with the parametric spontaneous down conversion during the oscillation build-up.
  • a portion of the signal from one degenerate OPO can be injected to the others.
  • Set the coupling strength ⁇ ⁇ 0.1 to represent a 26 dB loss.
  • the number of instances of the Ising model in the restricted set is equal to the number of cubic graphs.
  • N changes from 10 to 20
  • the computation time increases by about 8 times, much less than an exponential upsurge expected in a brute force search for the ground states.
  • the above theoretical investigation and numerical simulation demonstrate the potential for implementing the NP-hard Ising model on a coupled degenerate OPO network.
  • Two properties of the network are: the binary output phase of each degenerate OPO makes it possible to represent an Ising spin; and the minimum gain principle is that the oscillation mode of the system is likely to be the one with minimum overall photon decay rate at the steady state. Under weak mutual coupling among the degenerate OPOs, it has been shown that the mode with minimum overall photon decay rate corresponds to the ground spin configuration of the Ising model.
  • preliminary numerical simulations on instances in the Ising model that are equivalent to NP-hard MAX-CUT problems have demonstrated promising results for utilizing the system as an Ising machine.
  • results show the possibility of a new classical heuristic technique which solves the NP-hard Ising model using a set of nonlinear differential equations.
  • the Ising model is a mathematical abstraction of spin glasses composed of frustrated spins which possess peculiar properties such as spin freezing, cusp in magnetic susceptibility, remanence, and hysteresis.
  • the Ising model also serves as a cost function for various combinatorial optimization problems in biology, medicine, wireless communications, artificial intelligence, and social network. These problems are commonly believed to be intractable in computer science, and so far no efficient classical and quantum technique is known for solving them.
  • This disclosure describes a novel computing machine based on non-equilibrium phase transition exhibited by a coherent network of degenerate OPOs, which is efficient in solving Ising problems and other computationally difficult problems.
  • the network When the network is gradually pumped from below to above the oscillation threshold, it oscillates either in one of the ground states of the Ising Hamiltonian which have the minimum photon loss rate, or in low excess energy excited states which have slightly higher photon loss rates.
  • the network may be realized in one implementation in a single OPO ring cavity with multiple trains of pulses (e.g., femtosecond pulses) and configurable mutual couplings; spins are represented with above-threshold binary phases of the OPO pulses.
  • the smallest nondeterministic polynomial time (NP)-hard Ising problem was programmed on the machine, and in 1000 runs of the machine no computational error was detected.
  • FIG. 10A illustrates the operation of a degenerate OPO for below and above oscillation threshold in the in-phase and quadrature-phase coordinates.
  • the signal field is a squeezed vacuum state, and by increasing the pump field it undergoes spontaneous symmetry breaking around the oscillation threshold.
  • Gradual pumping of the OPO network is used to find the ground states (as indicated in FIG. 10B ).
  • the non-equilibrium phase transition leads to oscillation in one of the two phase states (
  • the couplings between the spins (J ij ) are realized by mutual injections of the signal fields of the i th and i th OPOs.
  • the resulting OPO network has a phase-state dependent photon decay rate corresponding to the energy landscape of the original Ising Hamiltonian; different spin configurations (phase states) have different total photon losses in the network, which is illustrated in FIG. 10C .
  • the search for the ground states is conducted by gradually raising the gain via increasing the pump field.
  • the network goes through the OPO phase transition, and as a result it is expected to oscillate in one of the exact or approximate ground states.
  • Further increase in the pump pins the gain to the threshold value and thus erroneous oscillation in the excited states continues to be suppressed.
  • the computational power of the OPO network resides in the transition from below to above the oscillation threshold.
  • This disclosure describes a numerical model using c-number Langevin equations, which are compatible with fundamental quantum mechanical master equations or Fokker-Planck equations, and the simulation results promise effectiveness of OPO networks in solving relatively large MAX-CUT problems and other difficult problems.
  • FIG. 11A illustrates the experimental setup using a 4-OPO system.
  • Ring resonator 1110 has a roundtrip time (T cavity ) of four times the pump pulse repetition period (T R ), as shown in FIG. 11B , and therefore the pulses represent four temporally separated independent OPOs.
  • T cavity roundtrip time
  • T R pump pulse repetition period
  • the couplings between these independent but identical OPOs are realized using three pairs of output and input couplers (OC/IC pairs 1120 , as shown in FIG. 11A ) in the resonator paths.
  • each delay is an integer multiple of the repetition period, and approximately 4% of the out-coupled light is injected back to the resonator. This corresponds to approximately 4% of field coupling between the OPOs.
  • FIG. 11C depicts how the delay lines provide all the possible couplings (J ij ) among the four temporally separated OPOs. Each delay line provides four couplings, i.e. two-body interactions, among the temporally separated OPOs.
  • Delay 1 couples OPO n to OPO n+1 (J 12 , J 23 , J 34 , J 41 ), delay 2 couples OPO n to OPO n+2 (J 13 , J 24 , J 31 , J 42 ), and delay 3 couples OPO n to OPO n+3 (J 14 , J 21 , J 32 , J 43 ).
  • the output was sent to a one-bit delay Michelson interferometer 1130 to measure the phase states of the 4-OPO system, as shown in FIG. 11A .
  • the delay difference of the arms is locked to the repetition period of the system (T R ). Therefore, the interferometer measures the differential phase between adjacent OPOs, such that the output is the interference of OPO n and OPO n+1 , and n changes with the time slot of measurement.
  • Fast and slow detectors were used at the output of the interferometer.
  • the 4-OPO system in the experimental setup was pumped with a femtosecond mode locked laser with a central wavelength of 1045 nm, at a repetition period of 4 ns.
  • the pump field was gradually increased from below the oscillation threshold to about two times the oscillation threshold over a time of approximately 180 ⁇ s, while the cavity photon lifetime is approximately 60 ns.
  • FIGS. 12A-C present plots from the slow detector output.
  • the OPOs are expected to operate independently, and therefore a random uniformly distributed selection of the phase states are expected each time the OPO is restarted.
  • the output toggles between the three distinct intensity levels (0, I m /2, and I m ). Assuming all the states are equally probable to occur, I m /2 corresponds to 12 phase states and I m , and 0 correspond to 2 different phase states, for a ratio of 6:1:1. As seen by the plot in FIG. 12A for all couplings blocked, the frequencies of occurrence of the 0, I m /2, and I m , events are 6:1:1.
  • in-phase (or out-of-phase) couplings among the adjacent OPOs corresponds to ferromagnetic (or antiferromagnetic) couplings in an Ising spin ring with the expected outcome of aligned (or anti-aligned) spins.
  • the shortest delay line was set to in-phase (or out-of-phase) coupling, and the other delay lines blocked.
  • the OPOs operated at the same phase states, whereas for the out-of-phase case, the OPOs operated in alternating states.
  • FIG. 12B illustrates the shortest delay line being swept through 360 degrees of phase with the other delay lines blocked. The sweeping speed is slow compared with the 1 kHz restarting frequency of the OPOs.
  • the phase of delay 1 is around 180°, such that the field of OPO n is shifted by ⁇ and injected to OPO n+1 , the output intensity remains at 0 level, meaning that the OPOs are either in
  • the coupled OPOs tolerate a coupling phase deviation of at least ⁇ 30° around in-phase or out-of-phase couplings. This regenerative behavior is due to the phase sensitive nature of the parametric gain in the degenerate OPO and makes the system nearly immune to the phase noise in the environment. Similar behavior was observed for the other delay lines.
  • the experimental setup was used to implement an NP-hard MAX-CUT problem, which is mapped to an Ising problem and corresponds to a frustrated Ising spin system.
  • the problem is to find a subset of vertices in a cubic graph, in which all vertices have three edges, such that the number of edges between the subset and its complementary subset is maximized.
  • any subset choice with two vertices is an answer of the MAX-CUT problem (or MAX-2-SAT problem), and any other subset choice is not.
  • the mutual couplings between the OPOs were set to out-of-phase (i.e. J ij ⁇ 0).
  • the total photon decay rate of the network is lowest for the answers of the MAX-CUT problem corresponding to two OPOs in
  • the phases of the three delays were locked to ⁇ using the residual pump interference signal at the transmission port of the input couplers.
  • a plot of the interferometer output is shown in FIG. 12C . It matches the expected outcome of being either I m /2 or 0 with the rate of I m /2 twice the rate of 0. However, since I m /2 corresponds to both answer and non-answer states, it is necessary to use a fast detector to confirm that the non-answer states do not occur.
  • a histogram of the eight phase states entered is shown in FIG. 13B .
  • the pulse train was detected and the corresponding phase state was inferred.
  • the result indicates uniform distribution of the phase states, confirming that four temporally separated OPOs are operating independently in the same resonator.
  • the histogram of the OPO states is measured for the case of all the delays locked to the phase of pi for 1000 trials.
  • the OPO network represents the MAX-CUT problem for a 4-vertex graph, and it only oscillates in the phase states corresponding to the answers of this problem. The result is depicted in FIG. 13C . No excited phase state is detected, and the distribution of the answer states is close to uniform. The error rate of computation is less than 10 ⁇ 3 which is limited by the length of measurements.
  • each phase state on the horizontal axis represents two complementary states, for example
  • No time reference is used in the measurements, and only the pattern of the pulses are used to infer the phase states: the measured entries for
  • 00 ⁇ are the number of [1010] patterns divided by 2.
  • the practical scalability enabled by multi-pulse operation in a long fiber-based ring cavity, simple mapping of the OPO bi-stable phase states to the Ising spin, and interesting quantum phase transition at the threshold can make the OPO network suitable for obtaining approximate solutions for large-scale NP-hard Ising problems with reasonable accuracy and speed.
  • intensity modulators are used to turn on and off each delay
  • phase modulators used to choose either in-phase or out-of-phase couplings. Exploiting optical fiber technologies and planar light wave circuits can enable implementation of compact large Ising machines.
  • the computational concept of this disclosure is compared with classical and quantum annealing techniques in the illustration in FIG. 14 , which illustrates the challenge in classical annealing to go from a metastable excited state to the ground state through multiple thermal hopping states, or the challenge in quantum annealing to go through quantum mechanical tunneling through many metastable states.
  • Classical simulated annealing employs a downward vertical search, in which the temperature is repeatedly decreased and increased until the ground state is found.
  • Quantum annealing exerts a horizontal search in the energy landscape with quantum tunneling. Therefore, with these methods, the computational time of finding the ground state increases with the increase in the number of metastable excited states or local minima.
  • the OPO-based Ising machine searches for the ground state in an upward direction.
  • the total energy i.e., the ordinate in FIG. 14
  • the optimum solution ground state has a minimum loss.
  • a parametric gain ‘G’ is included into the network and increased gradually, and the first touch to the network loss happens at the ground state, which results in the single-mode oscillation of the ground state spin configuration.
  • the OPO machine is a “heating machine” while the classical simulated annealing is a “cooling machine.” Since there is no structure below the ground state, as shown in FIG. 14 , the heating machine has some advantage over the cooling machine, and can be less susceptible to being trapped in the metastable excited states of an Ising problem.
  • , of the density matrix, ⁇ s and ⁇ p are the signal and pump photon decay rates,
  • Equation (27) can be cast into the c-number Langevin equation (C-LGE) for the in-phase and quadrature-phase components of the signal field via the Kramers-Moyal expansion, as shown in equations (30), (31).
  • a s1 2 A s 2 ⁇ ( a s +b s ) 2 +1 ⁇ /4 ⁇ A s1 2 (32)
  • ⁇ A s2 2 ⁇ A s 2 ⁇ ( a s ⁇ b s ) 2 ⁇ 1 ⁇ /4 ⁇ A s2 2 (33)
  • ⁇ A s1 2 A s 2 ( c 2 ⁇ c 2 (34)
  • ⁇ A s2 2 A s 2 ( s 2 ⁇ s 2 ) (35)
  • the Q-FPE and C-LGE for an injection-locked laser oscillator can be extended to the mutually coupled degenerate OPOs using equation (27) or equations (30), (31).
  • the resulting C-LGEs for a network of degenerate OPOs are given by equations (36), (37).
  • Table 3 summarizes performance of the OPO network in solving MAX-CUT problems on cubic graphs.
  • the success probability at the optimum pump rate for the worst instance is independent of the graph order and ranges from approximately 0.7-1.0.
  • V is the number of vertices in the graph
  • E is the number of edges
  • U SDP is the optimal solution to the semidefinite relaxation of the MAX-CUT problem
  • T is the average computation time of the OPO network normalized to the cavity photon lifetime.
  • both the best and average outputs of the OPO network are about 2-6% better than the 0.878-performance guarantee of the celebrated Goemans-Williamson algorithm based on semidefinite programming (SDP). Since the differences between the best and the average values are within 1% for most of the instances, reasonable performance is expected for the OPO network even in a single run, which makes the OPO network favorable for applications when response time is the utmost priority. In addition, there is further room to improve the performance, for example by applying local improvement to the raw outcomes of the OPO network and operating the OPO network under optimum pump rate of p and coupling strength of ⁇ .
  • the average computational time of the OPO network in solving the MAX-CUT problem on the worst-case cubic and G-set graphs is displayed in FIG. 16 .
  • the growth of the computation time fits well to a sub-linear function O(N 0.2 ).
  • This computational time is plotted in FIG. 16 by line 20 .
  • This general computational time for SDP is shown by line 30 in FIG. 16 . Since the OPO network is applicable to all types of graphs, the sub-linear scaling of the computation time gives it a huge advantage over the SDP algorithm in solving large-scale instances.
  • This computational time is compared to the time required to run the SDP (interior-point method) using a 1.7 GHz core i7 machine of about 1 ⁇ 10 5 sec, which is seven orders of magnitude larger than the approximately 1 ⁇ 10 ⁇ 2 sec for the OPO network.
  • Table 5 shows the phase states and corresponding output pulse trains in the first and second columns. Since the complementary phase states result in the same output, for the 16 possible phase states, 8 different pulse trains can occur. In the measurements with a fast detector, no time reference is used, therefore only four different pulse patterns can be detected. If a slow detector is used, the output will be proportional to the number of ones in the pulse train. Therefore, there will be three distinct output average intensity levels; namely Im, Im/2, and 0, as shown in the third column of Table 5. All 16 possible phase states of the 4-OPO system are shown in Table 5, with the corresponding pulse trains at the output of the unequal-arm interferometer, and the detected intensity using a slow detector.
  • the ring resonator 1110 of the OPO illustrated in FIG. 11A has a round trip time of 16 ns (a perimeter of about 4.8 m).
  • the actual setup included two more flat mirrors than shown in FIG. 11A (corresponding to a folded bow tie configuration).
  • the flat mirrors, except M 1 are gold coated with enhancement dielectric coatings at 2 ⁇ m.
  • One of the flat gold mirrors is placed on a translation stage with piezoelectric actuator (PZT).
  • the dielectric mirror (M 1 ) has a coating which is antireflective at the pump wavelength with less than 0.2% reflection, and is highly reflective (about 99%) from 1.8 ⁇ m to 2.4 ⁇ m.
  • the curved mirrors (M 3 and M 4 ) have 50-mm radius of curvature and are unprotected gold coated mirrors.
  • the angle of incidence on these mirrors is 4°, which was chosen to compensate the astigmatism introduced by the Brewster-cut nonlinear crystal, resulting in about 1 mm of cavity stability range for the spacing between the curved mirrors.
  • the signal beam has a waist radius of 8.3 ⁇ m (1/e 2 intensity) at the center of the crystal.
  • a 1-mm long, Brewster-cut, MgO:PPLN crystal has a poling period of 31.254 ⁇ m, which is designed to provide degenerate parametric gain for a pump at 1035 nm with type 0 phase matching (e ⁇ e+e) at 373 K temperature.
  • the crystal operates at room temperature in the OPO, and even though the phase matching condition is not optimal for the pump (centered at 1045 nm), degenerate operation is achieved by length tuning of the cavity.
  • Input and output coupling of the signal are achieved with 2- ⁇ m thick nitrocellulose pellicles to avoid dispersion in the cavity and etalon effects.
  • the three pairs 1120 of “OC” and “IC” are uncoated pellicles (with 2-6% power reflection), the “OC” for the main output is coated (with 15% power reflection at 2090 nm).
  • the beam splitter in the interferometer (BS) is the same coated pellicle.
  • another uncoated pellicle is used as an output coupler in the resonator (not shown).
  • the pump is a free-running mode-locked Yb-doped fiber laser (Menlo Systems Orange) producing approximately 80 fs pulses centered at 1045 nm with a repetition rate of 250 MHz, and maximum average power of >1 W.
  • the filter is a long pass filter at 1850 nm on a Ge substrate to eliminate the pump and transmit the signal.
  • Gradual pumping is achieved by the chopper, as it causes a rise time (10-90% power) of 180 ⁇ s for introducing the pump.
  • the cavity photon lifetime for the signal is estimated to be 60 ns, and the network is pumped approximately 2.2 times above threshold.
  • the controllers were used to stabilize the length of the cavity, the phase of the delay lines, and the arm-length difference of the interferometer.
  • the controllers are based on a “dither-and-lock” scheme, where a slight modulation (less than 10 nm amplitude at a frequency between 5 and 20 kHz) is applied to a fast PZT, and the error signal is generated electronically by mixing the detector output and the modulated signal.
  • Identical electronic circuits are used with a controller 3-dB bandwidth of 10 Hz.
  • the interference of the pump at the other port of the input couplers is used as the input of the controller, and the controller locks the length to achieve destructive interference on the detector, which results in constructive interference on the other port that enters the cavity.
  • the arm length difference of the interferometer is also locked similarly. No phase stabilization is required for the path from the OPO to the interferometer since all the OPO pulses experience the same path and phase change.
  • the servo controllers used in the experiment suffice for implementation of the Ising machine, and no stabilization on the pump is required.
  • Slow changes (within the response time of the controller) in the carrier-envelop offset frequency (f CEO p ) and repetition rate (f R ) of the pump do not affect the operation of the Ising machine.
  • Smooth changes in f R of the pump is intrinsically transferred to the signal since signal pulses are generated from pump pulses.
  • the effects of changes in f CEO p on the Ising machine require taking into account the intrinsic phase locking of the degenerate OPO as well as the role of servo controllers.
  • the primary task of the servo controller of the OPO is to maximize the output power by matching the roundtrip phase in the resonator to the pulse to pulse phase slip of the pump ( ⁇ p ).
  • the pulse to pulse phase slip is related to f CEO p by
  • phase slip of the signal pulses is locked to the phase slip of the pump with a factor of half, which consequently means the f CEO of the pump and signal are locked; the servo loop provides feedback to the cavity to follow this phase slip and maximize the output power.
  • a similar behavior also happens for a doubly-resonant OPO operating away from degeneracy, with a different ratio between the f CEO of the pump and signal.
  • the pulse to pulse phase slip in the pump transfers to the OPO to OPO phase slip by a factor of half Changing phase state from one OPO to another simply means adding it to the phase slip.
  • Locking a delay line to top of the fringe of pump pulses corresponds to having either 0 or ⁇ phase change for the signal. This is also true for the interferometer. In the experiments, for all the servo controllers, it was possible to precisely tune the length from one fringe to the next. This provided the ability to try different configurations and find the desired coupling phases.
  • the OPO When no output coupler is used in the cavity, the OPO has a threshold of 6 mW of pump average power. With all the output and input couplers in the cavity, the threshold reaches 135 mW. Oscillation at degeneracy and away from degeneracy can be achieved depending on the cavity length.
  • the OPO is pumped with 290 mW and the main output of the OPO has 15 mW of average power at degeneracy centered at 2090 nm with the spectrum shown in FIG. 17A .
  • the interferometric autocorrelation of the signal pulses is shown in FIG. 17B , suggesting a pulse length of approximately 85 fs.
  • the spatial profile of the output beam is very close to Gaussian as shown in FIG. 17C , with a radius of about 1 mm (at 1/e 2 intensity).
  • the average power of the signal in the delay lines are approximately 2 mW, and the intracavity power is estimated to be approximately 100 mW.
  • FIGS. 18A-F present results obtained using a slow detector for different combinations of couplings.
  • FIGS. 18A-C are obtained by scanning the phase of one delay line while the other delay lines are blocked.
  • Delay 1 and 3 have similar effects, because they couple adjacent OPOs but in different directions (see FIG. 11A ).
  • in-phase coupling by these delays results in the same phase state for all OPOs, and consequently high-intensity interferometer output (I m ); and out-of-phase coupling results in alternating phase states and consequently low-intensity interferometer output (0).
  • FIG. 18B the interferometer output is shown while the phase of delay 2 is scanned.
  • OPO 1 and 3 have the same phase state
  • OPO 2 and 4 oscillate in the same phase state.
  • these two pairs can either be the same or different, and therefore the output would be either I m or 0, as shown in FIG. 18B around phase of zero.
  • out-of-phase coupling of delay 2 results in constant output I m /2 as shown in the same plot. Regenerative behavior of the OPO and its insensitivity to a wide range of phase change in the couplings are observed in these three plots.
  • the phase of the delays was scanned one by one, and the results shown in FIGS. 19A-19C .
  • Different delay phase configurations and the expected outcomes are shown in Table 6, where the last row corresponds to the MAX-CUT problem with all anti-ferromagnetic couplings, and for each of the other rows one of the delay phases is different.
  • the center of the plot, where the phase of the scanned delay is ⁇ corresponds to the anti-ferromagnetic MAX-CUT problem.
  • the outputs follow the expected outcome.
  • Time division multiplexing facilitates scalability of the OPO network, and it benefits from intrinsically identical nodes in the network. It is also worth noting that for an Ising problem with N sites, the number of possible couplings (J ij ) is N 2 ⁇ N. However, in the time-division-multiplexed network only N ⁇ 1 delay lines are required for realizing these couplings, which means that the physical size of the machine scales linearly with N.
  • T R is the pulse-to-pulse interval
  • FIG. 20 A fiber-based implementation of such a machine is illustrated in FIG. 20 .
  • Using picosecond pump pulses enables implementation of a long resonator and long delay lines comprising optical fiber components.
  • T R 100 ps
  • a resonator with 200 m of optical fiber results in 10000 temporally separated OPOs.
  • the main challenge is stabilizing the phases of the fiber links
  • Use of low-noise phase-stabilized long (about 100-km) optical fibers can overcome this challenge.
  • the regenerative behavior suggests that the OPO-based Ising machine can tolerate relatively large phase noise in the couplings.
  • each delay line provides a delay of equal to an integer multiple of the repetition period (mT R ), and is responsible for multiple of the Ising coupling terms in the form of J (i)(i+m) .
  • each of these couplings happens at one time slot, and one can use electro-optic phase and amplitude modulators (EOM) to synchronously switch the delay on and off depending on whether the corresponding coupling term is zero or not.
  • EOM electro-optic phase and amplitude modulators
  • the computing machine uses time division multiplexing of optical pulses, in a network of OPOs implemented in an optical cavity or in optical fibers (or a combination of an optical cavity and optical fibers).
  • the pump energy may be of constant amplitude, or may be of increasing amplitude starting below an oscillation threshold.
  • Optical couplers are used to couple the OPOs together through delay components. The couplers may be introduced abruptly or gradually, and may be introduced before or after pumping begins.
  • the phase states of the OPOs are measured, and the phase states alone or in combination indicate a solution to a computational problem.
  • the measurement may be performed below or above the oscillation threshold (i.e., a quantum or a classical measurement, respectively.)
  • system control such as pump control, optical coupling control, and mirror control.
  • the pump frequency, ramp rate, amplitude, gain and other parameters may be controlled, and the optical coupling devices may be controlled to be on or off, and for abrupt or gradual change between on and off.
  • reflective surfaces in an optical cavity or in an interferometer may be controlled to set or fine-tune path length or configuration. Control of the optical couplings and control of the reflective surfaces may be individual, in banks, or all together. By way of example, optical coupling devices may be controlled individually, or as a group. Control of these various devices may be performed by one or more computing devices, as indicated in FIG.
  • Each computing device includes a processing device executing instructions.
  • the processing device may be one or more of a processor, microprocessor, microcontroller, ASIC, and/or FPGA, along with associated logic.
  • the instructions may be coded in hardware, or coded in firmware or software, or coded in some combination of hardware, firmware, or software.
  • Firmware and software may be stored in a memory.
  • Memory may be one or both of volatile and non-volatile memory for storing information. Examples of memory include semiconductor memory devices such as EPROM, EEPROM and flash memory devices, magnetic disks such as internal hard disks or removable disks, magneto-optical disks, CD-ROM and DVD-ROM disks, and the like.
  • An embodiment of the disclosure relates to a non-transitory computer-readable storage medium having computer code thereon for performing various computer-implemented operations.
  • the term “computer-readable storage medium” is used herein to include any medium that is capable of storing or encoding a sequence of instructions or computer codes for performing the operations, methodologies, and techniques described herein.
  • the media and computer code may be those specially designed and constructed for the purposes of the disclosure, or they may be of the kind well known and available to those having skill in the computer software arts.
  • Examples of computer-readable storage media include, but are not limited to: magnetic media such as hard disks, floppy disks, and magnetic tape; optical media such as CD-ROMs and holographic devices; magneto-optical media such as optical disks; and hardware devices that are specially configured to store and execute program code, such as ASICs, programmable logic devices (PLDs), and ROM and RAM devices.
  • magnetic media such as hard disks, floppy disks, and magnetic tape
  • optical media such as CD-ROMs and holographic devices
  • magneto-optical media such as optical disks
  • hardware devices that are specially configured to store and execute program code such as ASICs, programmable logic devices (PLDs), and ROM and RAM devices.
  • Examples of computer code include machine code, such as produced by a compiler, and files containing higher-level code that are executed by a computer using an interpreter or a compiler.
  • an embodiment of the disclosure may be implemented using Java, C++, or other object-oriented programming language and development tools. Additional examples of computer code include encrypted code and compressed code.
  • an embodiment of the disclosure may be downloaded as a computer program product, which may be transferred from a remote computer (e.g., a server computer) to a requesting computer (e.g., a client computer or a different server computer) via a transmission channel.
  • a remote computer e.g., a server computer
  • a requesting computer e.g., a client computer or a different server computer
  • Another embodiment of the disclosure may be implemented in hardwired circuitry in place of, or in combination with, machine-executable software instructions.
  • the terms “substantially”, “approximately” and “about” are used to describe and account for small variations.
  • the terms can refer to instances in which the event or circumstance occurs precisely as well as instances in which the event or circumstance occurs to a close approximation.
  • the terms can refer to less than or equal to ⁇ 10%, such as less than or equal to ⁇ 5%, less than or equal to ⁇ 4%, less than or equal to ⁇ 3%, less than or equal to ⁇ 2%, less than or equal to ⁇ 1%, less than or equal to ⁇ 0.5%, less than or equal to ⁇ 0.1%, or less than or equal to ⁇ 0.05%.

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